Absolutely continuous curves in Wasserstein spaces with applications to continuity equation and to nonlinear diffusion equations

Transcription

1 Università degli Studi di Pavia Dipartimento di Matematica F. Casorati Dottorato di ricerca in Matematica e Statistica Absolutely continuous curves in Wasserstein spaces with applications to continuity equation and to nonlinear diffusion equations Stefano Lisini Relatore Prof. Giuseppe Savaré Tesi di Dottorato - XVIII Ciclo

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3 III Contents Introduction 1 Acknowledgements 14 1 Preliminaries Absolutely continuous curves in metric spaces and metric derivative L p (I; X) spaces Metric Sobolev spaces W 1,p (I; X) Borel probability measures, narrow topology and tightness Push forward of measures Convergence in measure Disintegration theorem Kantorovitch-Rubinstein-Wasserstein distance Absolutely continuous curves in Banach Spaces Bochner and weak-* integral Radon-Nicodým property Characterization of absolutely continuous curves in Banach spaces Curves of maximal slope in metric spaces The minimizing movements approximation scheme Absolutely continuous curves in Wasserstein spaces and geodesics Characterization of the class of curves AC p (I; P p (X)) Wasserstein geodesics Length and geodesics spaces Characterization of geodesics of P p (X) The continuity equation The continuity equation in Banach spaces

4 IV CONTENTS A probabilistic representation of solutions of the continuity equation on R n The continuity equation in Banach spaces: results The continuity equation in R n with a Riemannian metric The Riemannian metric on R n The continuity equation in R n G Variable coefficients diffusion equations and gradient flows in Internal and Potential energy functionals Diffusion equations with variable coefficients Wasserstein contraction Stability of flows and convergence of iterated transport maps Stability of flows associated to a minimal vector field Flows and their stability On the regularity of the limit vectorfield Gradient flows of functional without geodesic convexity Subdifferential formulation of Gradient flows in P 2 () The gradient flow of the internal energy fuctional in Convergence of iterated transport maps

5 1 Introduction In this PhD Thesis we deal with some aspects of the applications of the theory of optimal mass transportation (and the related Kantorowitch-Rubinstein-Wasserstein distance) to evolution partial differential equations. Essentially, the Thesis is made up of three parts. The first part (Chapters 2 and 3) is devoted to the link between the measure valued solutions of the continuity equation, their probabilistic representation formula, and the absolutely continuous curves with values in the Wasserstein space P p (X). In particular, our main results (see [Lis5]) extend to an arbitrary separable and complete metric space X the previous contributions of [AGS5] (Hilbert spaces) and [LV5a] (locally compact metric spaces), and provide a characterization of the solutions of the continuity equation for a large class of Banach spaces. In the second part (Chapter 4) we apply the results obtained in the first part and the theory of gradient flows in metric spaces to the study of evolution equations of diffusion type with variable coefficients. We follow the interpretation of these equations, originally suggested by [JKO98], as gradient flows of suitable energy functionals in Wasserstein spaces. The third part (Chapter 5) is devoted to the problem of stability of flows associated to a sequence of non regular vector fields. We prove, in collaboration with Luigi Ambrosio and Giuseppe Savaré [ALS5], a general theorem of stability and we apply it to show the convergence of the iterated composition of optimal transport maps arising from the so called minimizing movements approximation scheme. Let us now explain in greater detail the main results of the present Thesis, referring to the single sections for more precise definitions, statements, and proofs of the results, as well as for further bibliographical notes. Continuity equation and absolutely continuous curves in P p (X). Continuity equation in R n. In order to illustrate the motivation that led us to the study of the absolutely continuous curves in Wasserstein spaces, we start with the principal

6 2 Introduction example: the continuity equation on R n t µ t + div(v t µ t ) =, in (, T ) R n. (1) Here µ t, t [, T ], is a family in P p (R n ), the set of Borel probability measures on R n with finite p-moment, i.e. x p dµ(x) < +, and v is a Borel velocity vector field R n v : (, T ) R n R n (we always use the notation v t (x) := v(t, x)) satisfying, for p > 1, the following integrability condition T R n v t (x) p dµ t (x) dt < +, (2) and the equation (1) has to be intended in the sense of distributions. We consider a continuous (in the sense of narrow topology, see Subsection 1.4) time dependent family of elements of P p (R n ), p > 1, µ t, t [, T ], which is a solution of (1). We assume, for a moment, that the vector field v is sufficiently regular, in such a way that, for every x R n, there exists a unique global solution of the Cauchy problem Ẋ t (x) = v t (X t (x)), X (x) = x, t [, T ]. (3) In this case, it is well known that the solution of equation (1) is representable by the formula µ t = (X t ) # µ. (4) The expression (X t ) # µ denotes the push forward of the initial measure µ through the map X t : R n R n, which is defined by (X t ) # µ (B) := µ ((X t ) 1 (B)) for every Borel set B of R n. Taking into account (3) and (2), and using Hölder s inequality, we obtain that for every s, t [, T ], with s < t, R n X s (x) X t (x) p dµ (x) (t s) p 1 t s R n v r (x) p dµ r (x) dr. Recalling the definition of the p-wasserstein distance between µ, ν P p (R n ), W p (µ, ν) := ( { }) 1 min x y p p dγ(x, y) : γ Γ(µ, ν), R n R n where Γ(µ, ν) is the set of Borel probability measures on the product space R n R n having first marginal equal to µ and second marginal equal to ν, and using the admissible measure γ = (X s ( ), X t ( )) # µ Γ(µ s, µ t ) we obtain the inequality t Wp p (µ s, µ t ) (t s) p 1 s R n v r (x) p dµ r (x) dr. (5) This last inequality and (2) imply that the curve t µ t is absolutely continuous in P p (R n ). By Lebesgue differentiation Theorem, we have that µ p W p p (µ s, µ t ) (t) := lim s t t s p v t (x) p dµ t (x) R n for a.e. t (, T ), (6)

7 Introduction 3 and, because of the assumption (2), we obtain that the curve µ t belongs to the space AC p ([, T ]; P p (R n )), i.e. the space of the absolutely continuous curves µ : [, T ] P p (R n ) such that µ L p (, T ). In the applications, it is important to work with non regular vector fields v, satisfying only (2). In this case, the flow X t associated to v is not defined, in general, and the representation (4) does not make sense. Nevertheless, another type of representation, strictly linked to the previous one, is possible (see Theorem 3.1): every continuous time dependent Borel probability solution t µ t of the continuity equation (1) with the vector field v satisfying (2) is representable by means of a Borel probability measure η on the space of continuous functions C([, T ]; R n ). The measure η is concentrated on the set of the curves {t X t : X is an integral solution of (3) and Ẋ L p (, T ; R n )}. Now the relation between η and µ t is given by (e t ) # η = µ t t [, T ], (7) where e t : C([, T ]; R n ) R n denotes the evaluation map, defined by e t (u( )) := u(t), and the push forward is defined by (e t ) # η(b) := η({u C([, T ]; R n ) : u(t) B}) for every Borel set B of R n. When (7) holds, we say that η represents the curve µ t. Starting from this representation of the solution µ t of (1) and taking into account the set where η is concentrated, it is not difficult to show that the curve µ t belongs to the space AC p ([, T ]; P p (R n )) and the estimate (6) still holds. Since, for a given curve t [, T ] µ t, solution of (1), there are many Borel vector fields v satisfying (2) such that (1) holds, a natural question arises: does a Borel vector field ṽ exist such that the continuity equation (1) holds, and ṽ is a minimizer of the L p norm T v R n t (x) p dµ t (x) dt? For p = 2 this problem has a natural physical interpretation: since the norm v R n t (x) 2 dµ t (x) is the kinetic energy at the time t, we are looking for a vector field v which minimizes the action T v R n t (x) 2 dµ t (x) dt. The answer is positive and the minimizers are characterized by the fact that the equality holds in (6). Moreover this minimizing vector field is unique and, thanks to the equality in (6), it plays the role of tangent vector to the curve t µ t. This result holds also for separable Hilbert spaces ([AGS5] Theorem 8.3.1). In this Thesis we extend it to a suitable class of separable Banach spaces, and to dual of separable Banach spaces, even considering solutions without finite p-moment; both these results are, in fact, a direct consequence of a more general property which holds in arbitrary separable and complete metric spaces and which is interesting by itself. Indeed, recently many papers have appeared dealing with various aspects of measure metric spaces or, in particular, of Riemannian manifolds, strictly connected to Kantorovitch-Rubinstein-Wasserstein distance

8 4 Introduction (see e.g. [LV5a], [LV5b], [Stu5b], [Stu5c], for metric measure spaces, and [WO5], [OV], [Stu5a], [vrs5], [CEMS1] for Riemannian manifolds). The case of an arbitrary separable and complete metric space. In a separable and complete metric space X, without additional structure, the notions of velocity vector field and of continuity equation do not make sense, but we still have the notion of (scalar) metric velocity of absolutely continuous curves, which is sufficient to obtain quantitative estimates like (6) as well as for the opposite one. We recall that, when u : [, T ] X is an absolutely continuous curve, for almost every t [, T ], the following limit u (t) := lim h d(u(t + h), u(t)) h exists, and u L 1 ([, T ]). We denote by AC p ([, T ]; X) the class of absolutely continuous curves, defined on [, T ] with values in X, such that u L p (, T ). We also recall that on P p (X), the set of Borel probability measures on X with finite moment of order p, the p-wasserstein distance (more properly the p-kantorowitch-rubinstein- Wasserstein distance) is defined by W p (µ, ν) := ( { }) 1 min d p p (x, y) dγ(x, y) : γ Γ(µ, ν), X X where Γ(µ, ν) is the set of Borel probability measures on the product space X X having first marginal equal to µ and second marginal equal to ν. P p (X) is a separable, complete metric space as well. Let us try to give a brief account of the metric point of view, describing the main results. In Theorem 2.2 we prove that, given a curve t µ t belonging to AC p ([, T ]; P p (X)), for p > 1, there exists a Borel probability measure η on the space C([, T ]; X) of the continuous curves in X, which is concentrated on the set AC p ([, T ]; X). This measure η represents the curve µ through the relation (e t ) # η = µ t t [, T ]. (8) Moreover, and most importantly, this measure plays the role of a minimal vector field. Indeed, the following inequality holds u p (t) d η(u) µ p (t) for a.e. t (, T ). (9) C([,T ];X) On the other hand, in Theorem 2.1 we prove that if η is a Borel probability measure on the space C([, T ]; X), concentrated on the set AC p ([, T ]; X) such that T C([,T ];X) u p (t) dt dη(u) < +, (1)

9 Introduction 5 then the curve defined by µ t := (e t ) # η belongs to AC p ([, T ]; P p (X)) and the opposite inequality holds µ p (t) u p (t) dη(u) for a.e. t (, T ). C([,T ];X) Then the equality holds in (9) and, consequently, the measure η satisfies a kind of minimality property. We observe that the condition (1) is the metric counterpart of the integrability condition (2). Notice that Theorem 2.2 is a theorem of representation of curves in AC p ([, T ]; P p (X)) as superposition of curves of the same kind, AC p ([, T ]; X), in the space X. In order to cover the various cases, which are important for the applications, we will also extend Theorem 2.2 in two directions. The first extension is given by Corollary 2.3 and deals with the case of a pseudo metric space (where the distance d can assume the value +, see Remark 1.2). An important example is provided by P(X), the space of Borel probability measures on X without assumptions on the finiteness of the p-moments, endowed with the p-wasserstein (pseudo)distance: it is a pseudo metric space when X is not bounded. This extension to pseudo metric spaces is possible since all the results of Theorems 2.1 and 2.2 are expressed in terms of the metric derivative, and this concept involves only the infinitesimal behaviour of the distance along the curve. Then we can substitute the pseudo distance with a topologically equivalent bounded distance and with the same infinitesimal behaviour. It is also interesting to note that if two (topologically equivalent) distances on the same space X induce the same class of absolutely continuous curves and the same metric velocity, then the corresponding Wasserstein pseudo distances on P(X) enjoy the same property (Corollary 2.4). The second easy extension, Corollary 2.6, concerns a possible application to the case of non separable metric spaces. Considering the set of tight measures (see Remark 1.11) it is not difficult to show that if all the measures µ t of a curve belonging to AC p ([, T ]; P(X)) are tight, then they are supported in a separable closed subspace of X. Geodesics in P p (X). A first application of Theorems 2.1 and 2.2 is given in Section 2.2 and provides a characterization of the geodesics of the metric space P p (X) as superposition of geodesics of the metric space X, under the hypothesis that X is a length space (i.e., for every couple x, y of points of X, the distance between x and y is the infimum of the lengths of absolutely continuous curves joining x to y). First of all we show that if X is a length space then P p (X) is a length space as well (Proposition 2.7). In Theorem 2.9 we prove the characterization of geodesics of P p (X) by a straightforward application of Theorem 2.2, since the geodesics are a particular class of absolutely continuous curves. A similar result was obtained in [LV5a] with the further assumption that the space X is locally compact

10 6 Introduction (see also [Vil6]). We point out that a characterization of the geodesics of the space P p (X) is also useful in order to prove convexity along geodesics of functionals defined on P p (X) (on this important subject see e.g. [McC97] where the convex functionals in P 2 (R n ) were studied for the first time, and [Stu5a], [LV5a] where the convexity is used in order to give a definition of Ricci curvature bounds in metric measure spaces). Continuity equation in Banach spaces. Chapter 3 is devoted to the deep link between solutions of the continuity equation with vector fields satisfying a suitable condition of L p integrability and curves of AC p ([, T ]; P p (X)). In Section 3.1 we study the continuity equation in Banach spaces. The continuity equation in a Banach space X t µ t + div(v t µ t ) = (11) where µ t, t [, T ], is a time dependent continuous family of probability measures in the space X and v : (, T ) X X is a vector field satisfying the L p integrability condition T v t (x) p dµ t (x) dt < +, (12) X is imposed in the duality with smooth functions with bounded (Frechét) differential (we refer to Section 3.1 for the precise notion of solution). In Theorem 3.2 we prove that in a separable Banach space X satisfying the Radon- Nicodým property (see Subsection 1.6.2) and in the dual of a separable Banach space, for any curve µ AC p ([, T ]; P p (X)), p > 1, there exists a vector field ṽ, satisfying (12) and the continuity equation, such that the following inequality (analogous of (9)) holds ṽ t (x) p dµ t (x) µ p (t) for a.e. t [, T ]. (13) X The proof of the existence of a minimal vector field is based on the representation Theorem 2.2 for curves AC p ([, T ]; P p (X)), and on the differentiability (in a strong vector sense) almost everywhere for absolutely continuous curves in Banach spaces having the Radon- Nicodým property. Thanks to these results, by disintegrating the measure η, representing the curve µ, with respect to e t (see Subsection for the disintegration Theorem) and denoting the disintegrated measures by η t,x, in Banach spaces having the Radon-Nicodým property it is possible to construct a vector field ṽ t (x) := u(t) d η t,x (u) (14) {u C([,T ];X):u(t)=x} satisfying the continuity equation and the inequality (13). The integral in (14) is a Bochner integral and we refer to the proof for the discussion about the good definition and the measurability of the vector field ṽ.

11 Introduction 7 With some modifications the proof works also for the dual of a separable Banach space. In this case the absolutely continuous curves, in general, are not differentiable almost everywhere with respect to the strong topology, but are differentiable almost everywhere with respect to the weak-* topology (see Theorem 1.16). However, this weak-* differentiability, together with the metric differentiability, is sufficient to carry out the proof, where now the measurability of a vector field has to be intended in the weak-* sense and the integral (14) is the weak-* integral (see Subsection 1.6.1). The next relevant question concerns the possibility to prove if equality holds in (13). As in the case of R n, if every continuous solution µ t of the continuity equation, with the vector field v satisfying (12), belongs to AC p ([, T ], P p (X)) and µ p (t) v t (x) p dµ t (x) for a.e. t [, T ], X then we can conclude that the equality holds in (13). In Theorem 3.6 we adopt the strategy of reducing the statement to the finite dimensional case by projecting X on spaces of finite dimension. A crucial assumption here is a property of approximation for the Banach space X, precisely the standard Bounded Approximation Property (see Definition 3.3), which can be suitably modified when X is the dual of a separable Banach space (see Definition 3.4). In Remark 3.5 we give some examples of Banach spaces satisfying these properties. Here we point out that we can apply our results to the space X = l = (l 1 ), which is particularly relevant since it contains an isometric copy of any separable metric space (see Remark 3.5). The final main result of Section 3.1 is then Theorem 3.7 which states that in a separable Banach space having Radon-Nicodým property and Bounded Approximation Property (or in the dual of a separable Banach space satisfying a weak-* Bounded Approximation Property) the vector field ṽ defined in (14) realizes the equality in (13) and is of minimal norm among all others vector fields such that the continuity equation and the integrability condition (12) hold. If the norm of X is strictly convex, then the minimal vector field is uniquely determined for µ t -a.e. x X and a.e. t [, T ]. As a final observation, we point out that under the assumptions that the separable Banach space X satisfies the Radon-Nicodým property and the Bounded Approximation Property (or X is the dual of a separable Banach space satisfying the weak-* Bounded Approximation Property), using the fact that P p (X) is a geodesic space (Proposition 2.7), Theorem 3.6 and the existence of a minimal vector field ṽ, we recover the Benamou-Brenier formula ([BB]) { 1 } Wp p (µ, ν) = min v t (x) p dµ t (x) dt : (µ t, v t ) A (µ, ν), (15) X where A (µ, ν) is the set of the couples (µ t, v t ) such that t µ t is (narrowly) continuous, µ = µ, µ 1 = ν, v t satisfies (12) with T = 1, and the continuity equation (11). In Corollary 2.1 we give also a metric version of this formula in geodesic metric spaces.

12 8 Introduction All the results of Chapter 2 and of Section 3.1 are contained in the paper [Lis5]. The continuity equation in R n with a non smooth Riemannian distance. Theorem 3.1 extends the validity of Theorem 3.7 to the case of a (non smooth) Riemannian distance on R n. Even if we assume that the distance is equivalent to the euclidean one, this is not a consequence of the previous result, since the Wasserstein distance depends on the metric of the space. More precisely, we consider the Riemannian distance { 1 } d(x, y) = inf G(γ(t)) γ(t), γ(t) dt : γ AC([, 1]; X), γ() = x, γ(1) = y, induced by a given metric tensor, represented by a symmetric matrix valued function G : R n M n satisfying a uniform ellipticity condition λ ξ 2 G(x)ξ, ξ Λ ξ 2 x R n ξ R n (16) for some constants Λ, λ >, and the following regularity property x G(x)ξ, ξ is lower semi continuous ξ R n. (17) Again, the proof is based on the metric Theorem 2.2 and the property of almost-everywhere differentiability of absolutely continuous curves. We observe that (17) has been assumed only to prove the equality (see Proposition 3.8) u (t) = G(u(t)) u(t), u(t) for a.e. t [, T ], (18) for any absolutely continuous curve u : [, T ] R n, when R n is endowed with the distance d. Diffusion equations with variable coefficients. A suitable Riemannian metric on R n arises naturally when we try to interpret the diffusion equations with variable coefficients as gradient flows of a suitable energy functional with respect to a Wasserstein distance. Chapter 4 is entirely devoted to this issue. In the case of the linear Fokker-Planck equation and of the porous media equation, the gradient flow structure with respect to the 2- Wasserstein distance has been pointed out by [JKO98] and [Ott1]. In [JKO98], the seminal paper on this subject, the authors prove that the solutions of the linear Fokker-Planck equation t u div( u + u V ) =, in (, + ) R n, (19) with initial datum u, u L 1 (R n ) = 1, x 2 u R n (x) dx < +, can be constructed as limits of the (variational formulation of the) Euler implicit time discretization of the gradient flow induced by the functional φ(u) := u(x) log(u(x)) dx + V (x)u(x) dx (2) R n R n

13 Introduction 9 on the set {u L 1 (R n ) : u, u 1 = 1, R n x 2 u(x) dx < + } (here we identify the probability measures with their densities with respect to the Lebesgue measure L n ) with respect to the 2-Wasserstein distance. Namely, given a time step τ > and an initial datum u, they consider the sequence (u k ) obtained by the recursive minimization of u 1 2τ W 2 2 (u, u k 1 ) + φ(u), k = 1, 2,, (21) with the initial condition u = u, and the related piecewise constant functions U τ,t := u [t/τ], t >, ([t/τ] denoting the integer part of t/τ) interpolating the values u k on a uniform grid {, τ, 2τ,, kτ, } of step size τ. The authors prove the convergence for τ of U τ to a solution of (19), when V is a smooth non negative function with the gradient satisfying a suitable condition of growth. This approximation scheme is the minimizing movements approximation scheme introduced in the general setting of metric spaces by De Giorgi [DG93]. For a larger class of equations, the problem of the convergence to the solution of the method described above has attracted a lot of attention in recent years. Related results are treated in [Ott96], [Agu5], [CG3], [CG4]; a comprehensive convergence scheme at the PDE level which encompasses the Otto s method is illustrated in of [AGS5]. Another interesting paper is [CMV6] where the interpretation of a class of evolution equation as gradient flow allows to deduce new properties of the solutions. Here we consider the class of nonlinear diffusion equations, with a drift term, of the type t u div( f(u) + u V ) =, in (, + ) R n, (22) with initial datum u, u L 1 (R n ), R n x 2 u (x) dx < +, where f is a suitable non decreasing function and V a sufficiently regular function. The equation (22) is the gradient flow (see e.g. [AGS5]) of the energy functional φ(u) := F (u(x)) dx + R n V (x)u(x) dx R n (23) defined on {u L 1 (R n ) : u, u 1 = 1, R n x 2 u(x) dx < + }, where F is a regular convex function, linked to f by the relation F (u)u F (u) = f(u). In this Thesis we consider the variable coefficients nonlinear diffusion equation with drift t u t (x) div(a(x)( f(u t (x)) + u t (x) V (x)) =, (24) where A is a symmetric positive definite matrix valued function, depending on the spatial variable x R n (we use always the notation u t (x) := u(t, x)). The equation (24) describes linear or nonlinear diffusion with drift in nonhomogeneous and anisotropic material. There are many references for diffusion equations in the fields of mathematical analysis, mathematical physics and of the applications. For instance we can

14 1 Introduction see [Cra75] for diffusion equations, [Ris84] and [Fra5] for linear and non linear Fokker- Planck equations. Diffusion equations with variable coefficients arises also in models of economy [BM] and in approximation of kinetic models [PT5], or in the linearization of fast diffusion equation [CLMT2]. Under the assumptions that the matrix function A is sufficiently regular (for instance of class C 2 ), that V is smooth, and f(u) = u, the proof of [JKO98] can be modified in order to show that (24) is the gradient flow of the functional (23) with respect to the 2-Wasserstein distance on R n, where R n is endowed with a suitable Riemannian distance d. Precisely the distance induced by the metric tensor G = A 1. When A is less regular, for instance only continuous, this kind of proof fails, even if the algorithm (21) is still meaningful. Our purpose is to show, also in the case of low regularity of A, that the minimizing movements algorithm for the functional (23) still converges to a solution of the equation (24). In order to show this, we follow the approach of the theory of curves of maximal slope in metric spaces ([DGMT8], [DMT85], [Amb95], [AGS5]). Combining that theory with our previous result (Theorem 3.1) on the existence of the minimal vector field, we obtain our main result about the diffusion equation with variable coefficients: Theorem 4.4. Here A : R n M n is a symmetric matrix valued function, Borel measurable, satisfying a uniform ellipticity condition λ ξ 2 A(x)ξ, ξ Λ ξ 2 x R n ξ R n, (25) such that the inverse matrix function G := A 1 satisfies (17), V : R n (, + ] is a convex lower semi continuous function, bounded from below, whose domain has non empty interior, and F : [, + ) (, + ] is a regular convex function satisfying suitable conditions, listed in Chapter 4. The assumptions on F and V ensure that the functional φ is convex along geodesics of P 2 (R n ) with the standard Wasserstein distance induced by the euclidean distance (notice that, in general φ is not convex along geodesics of P 2 (R n ) with the Wasserstein distance induced by the Riemannian distance d). Under these assumptions, the algorithm (21), starting from u in the domain of φ, converges (up to sub sequences) to a distributional solution u of the nonlinear diffusion equation (24). Given a bounded convex open set R n, we observe that the function if x V (x) = + if x R d \. is an admissible potential. Hence our theorem provides a solution of (24) in (, + ) satisfying (a weak formulation of) the homogeneous Neumann boundary condition A f(u t (x)) n(x) = in (, + ). A related issue is the long time asymptotic behavior for the equation (24). It is important to have conditions on the functional φ which ensure the convergence of the solution, as

15 Introduction 11 t, to the stationary state, when it exists and it is unique. The stationary state, corresponding to a minimum of the functional φ, when it exists and it is unique, is the same for the equation (24) and for the same equation with A = I. Applying a result of [CMV3] we can prove significant results of asymptotic behaviour for our equation with variable coefficients. When the potential function V is regular, non negative and strictly uniformly convex, i.e. satisfying 2 V (x) αi for α >, then there exists a unique minimizer u for the functional φ, which turns out to be a stationary state for the equation (24). In Theorem 4.5 we prove that there is an exponential rate of convergence for t in relative entropy, more precisely if u t is a solution of the equation, with initial datum u in the domain of φ, given by Theorem 4.4 then φ(u t ) φ(u ) e 2λαt (φ(u ) φ(u )) t (, + ), (26) where λ is the ellipticity constant in (25). Moreover there is also exponential rate of convergence in Wasserstein distance W 2,G (u t, u ) e λαt 2 λα (φ(u ) φ(u )) t (, + ). (27) In the particular case of linear diffusion, where F (u) = u log u, thanks to Csiszár-Kullback inequality u ũ 2 L 1 (R n ) 2(φ(u) φ(ũ)) (28) (see [Csi63], [Kul59] and [UAMT] for generalized version), we also have convergence in L 1 (R n ) with exponential rate of decay u t u L1 (R n ) e λαt 2(φ(u ) φ(u )) t (, + ). (29) Rate of convergence in various relative entropies, for the linear Fokker-Planck equation with variable coefficients, are studied in the interesting paper [AMTU1]. Another problem is that of the contractivity of the Wasserstein distance along the solutions of the equation. In recent years, for several classes of evolution equations, this issue has attracted a lot of attention, see e.g. [CMV6], [LT4], [vrs5] in Riemannian manifolds, and [AGS5] also in the abstract metric context. The problem can be described as follows: given two initial data u 1 and u 2 in the domain of the functional φ, and u 1, u 2 the corresponding solutions of the equation, does a constant α R exist such that Clearly the most interesting case is α <. W 2 (u 1 t, u 2 t ) e αt W 2 (u 1, u 2 ) t (, + )? (3)

16 12 Introduction About this problem, for variable coefficients equation, we have only a partial result. In Theorem 4.6 we prove the contraction inequality (3) only for the linear diffusion and a scalar matrix A(x) = a(x)i, under a suitable hypothesis on the coefficients and the potential (see the assumption (4.64)). All the results of this Chapter are contained in the paper [Lis6]. Convergence of iterated transport maps. In Chapter 5 (in collaboration with Luigi Ambrosio and Giuseppe Savaré, see [ALS5]) we deal with the problem of the convergence of iterated transport maps arising from the discretization method (21). In order to illustrating the problem, let us assume that we are given a functional φ defined on P 2 (R n ), whose domain is contained in the set of measures absolutely continuous with respect to the Lebesgue measure L n (for example, the internal energy functional (23)). Let us consider the minimizing movement approximation scheme for φ, with respect to the 2-Wasserstein distance W 2, introduced in (21). Under quite general assumptions on φ, as illustrated in Theorem 5.8, it is possible to prove the existence of the limit u t L n = lim τ U τ,t L n in P 2 (R n ) with respect to W 2, for every time t. The limit curve µ t = u t L n is a solution of the continuity equation for a vector field v t L 2 (µ t ; R n ) linked to µ t itself through a nonlinear relation depending on the particular form of the functional φ. When a suitable upper gradient inequality is satisfied by µ t, it turns out that v t is the minimal vector field as in Chapter 3. Denoting by t k the unique optimal transport map between µ k = u k L n and µ k+1 = u k+1 L n, (see Section 1.5), we consider the iterated transport map T k := t k 1 t k 2 t 1 t, mapping µ = µ to µ k. We want to study the convergence of the maps T τ,t := T [t/τ] as τ. A simple formal argument shows that their limit should be X t, where X t is the flow associated to the minimal vectorfield v t, i.e. a map defined for µ-a.e. x R n, as the solution of the Cauchy problem d dt X t(x) = v t (X t (x)), X (x) = x. In order to make this intuition precise, and to show the convergence result, we use several auxiliary results, all of them with an independent interest: the first one, proved in Section 5.1, is a general stability result for flows associated to vector fields in the same spirit of the results proved in [DL89], [Amb4b] and based in particular on the Young measure technique in the space of (absolutely) continuous maps adopted in [Amb4b] (see also the Lecture (31)

17 Introduction 13 Notes [Amb4a]). The main new feature here, with respect to the previous results, is that we use the information that the limit vector field v is the minimal vector field, while no regularity is required either on the approximating vector fields or on the approximating flows. More precisely the main stability result for flows, Theorem 5.1 can be illustrated as follows. Let (µ n t, v n t ) be a sequence of solutions of the continuity equation and let (µ t, v t ) be a solution of the continuity equation, such that the sequence µ n t narrowly converges to µ t and the sequence of the L p norms of v n t converges to the one of v t. We prove that the flows associated to the fields v n t converge (in a kind of L p sense) to the flow associated to v t, under the assumption that v t is the minimal vector field associated to µ t and the Cauchy problem (31) admits at most one solution for µ -a.e. x R n. Notice that we have not assumed any regularity on the approximating vector fields ensuring the uniqueness of the approximating flows, but only the existence of the flows. In Subsection we collect some sufficient conditions on the vector field v ensuring that the hypothesis of the stability Theorem are satisfied. In Section 5.2 we recall from [AGS5] some definitions about the subdifferential formulation of gradient flows of a functional φ in P 2 (R n ) and we prove the general Theorem 5.8 relative to the convergence of the discrete solutions M τ to the continuous one µ. In Proposition 5.9 we show that the convergence scheme works for the gradient flow of the internal energy functional φ(µ) := F (u) dx, with µ = ul d, (32) under the assumption of boundedness of the initial density u. We point out that we have not required conditions ensuring the geodesic convexity of φ. Namely we have not required the convexity of and the McCann displacement convexity condition on F. The crucial role in the proof is played by an upper gradient inequality. In this case the gradient flow of φ corresponds to a weak formulation of a suitable nonlinear diffusion equation with homogeneous Neumann boundary conditions. In the final Section 5.3, in Theorem 5.13, we prove the convergence, as τ, of the iterated transport map T τ,t, previously defined, to the flow X associated to v when the Cauchy problem (31), admits at most one solution for µ -a.e. x R n. The proof is an application of the stability result in Theorem 5.1. Finally, in order to the apply Theorem 5.13 to the internal energy functional (the main concrete example) we discuss some sufficient conditions, depending on the regularity of the initial data. This discussion is summarized in Corollary In the paper [ALS5] the results about the convergence of the iterated transport map T τ are applied to the study of the gradient flow in L 2 (, R n ) of a particular class of policonvex functionals, solving an open problem of convergence raised in the beautiful paper [ESG5].

18 14 Introduction Acknowledgements I am deeply indebted to my PhD advisor, Professor Giuseppe Savaré for suggesting me this interesting research subject, for sharing his time with me and for the help he always gave me. I would also like to thank Professor Luigi Ambrosio for many useful suggestions and for his kind collaboration. I would like to thank Professor Giuseppe Toscani for some discussions on the subject of the thesis. Finally I would like to thank the anonymous referee for the useful observations and the interesting suggestions for a future development of my research.

19 15 Chapter 1 Preliminaries In this Chapter we define the main objects we are dealing with and we state some well known results (in general without proof) which are important in the development of the Thesis. In all this Chapter we denote by I the bounded, closed interval [, T ]. 1.1 Absolutely continuous curves in metric spaces and metric derivative In this Section we give the definition of the class of absolutely continuous curves with values in a general metric space, and we recall a basic result about metric differentiability. Let (Y, d) be a metric space. We say that a curve u : I Y belongs to AC p (I; Y ), p 1, if there exists m L p (I) such that d(u(s), u(t)) t s m(r) dr s, t I s t. (1.1) A curve u AC 1 (I; Y ) is called absolutely continuous in Y, and a curve u AC p (I; Y ), for p > 1, is called absolutely continuous with finite p-energy. The elements of AC p (I; Y ) satisfy the nice property of a.e. metric differentiability. Precisely we have the following Theorem (see [AGS5] for the proof). Theorem 1.1. If u AC p (I; Y ), p 1, then for L 1 -a.e. t I there exists the limit lim h d(u(t + h), u(t)). (1.2) h We denote the value of this limit by u (t) and we call it metric derivative of u at the point t. The function t u (t) belongs to L p (I) and d(u(s), u(t)) t s u (r) dr s, t I s t. Moreover u (t) m(t) for L 1 -a.e. t I, for each m such that (1.1) holds.

20 Preliminaries Remark 1.2 (pseudo distance). If d : Y Y [, + ] satisfies all the usual axioms of the distance but can also assume the value +, we call it pseudo distance 1 and the space (Y, d) pseudo metric space. A pseudo distance induces on Y a topology (the topology generated by the open balls) exactly as a distance and, defining d(x, y) := d(x, y) 1, the space Ỹ := (Y, d) is a bounded metric space, topologically equivalent to Y (see [Bou58]). We observe explicitly that C(I; Y ) = C(I; Ỹ ), the definition of absolutely continuous curves in Y makes sense and Theorem 1.1 holds. Moreover, if u AC p (I; Y ) then d(u(t), u(s)) < + for every s, t I, and the metric derivative of u with respect to d coincides with the metric derivative of u with respect to d. Then it follows that AC p (I; Y ) = AC p (I; Ỹ ). 1.2 L p (I; X) spaces In this Section we introduce the metric valued L p (I; X) spaces and we recall an important compactness criterion in L p (I; X), proved in [RS3], which is a key tool in the proof of our main result in Chapter 2. Let (X, d) be a separable, complete metric space. We say that a function u : I X belongs to L p (I; X), p [1, + ) if u is Lebesgue measurable and T for some (and thus for every) x X. d p (u(t), x) dt < + The metric space L p (I; X) is the space of equivalence class (with respect to the equality a.e.) of functions in L p (I; X), endowed with the distance ( T d p (u, v) := d p (u(t), v(t)) dt Since X is separable and complete, the space L p (I; X) is separable and complete too. We recall a useful compactness criterion in L p (I; X) (it follows from Theorem 2, Proposition 1.1, Remark 1.11 of [RS3], since the last two can be extended to L p (I; X)). Theorem 1.3. A family A L p (I; X) is precompact if A is bounded, lim sup h u A T h ) 1 p d p (u(t + h), u(t)) dt =, 1 Often pseudo distance means a d : Y Y [, + ] which satisfies all the usual axioms of the distance, except the property d(x, y) = y = x. In place of this property, it satisfies only d(x, x) =. We do not consider this variant. In [Bou58] the application d is called écart..

21 1.3 - Metric Sobolev spaces W 1,p (I; X) 17 and there exists a function ψ : X [, + ] whose sublevels λ c (ψ) := {x X : ψ(x) c} are compact for every c, such that sup u A T ψ(u(t)) dt < +. (1.3) 1.3 Metric Sobolev spaces W 1,p (I; X) In this Section we give the definition of W 1,p (I; X) Sobolev spaces with values in the metric space X. In the finite dimensional case X = R n it is well known that the Sobolev spaces W 1,p (I; R n ), for p > 1 can be characterized by { u L p (I; R n ) : sup <h<t T h where h u, for h (, T ), denotes the differential quotient h u(t) := ( h u(t)) p dt < + u(t + h) u(t), t [, T h]. h When X is a separable, complete metric space and p > 1, still denoting by h u, for h (, T ), the differential quotient we can define W 1,p (I; X) := h u(t) := { d(u(t + h), u(t)), t [, T h], h u L p (I; X) : sup <h<t T h } ( h u(t)) p dt < +, }. (1.4) The following Lemma shows that the spaces AC p (I; X) are strictly linked to the Sobolev spaces W 1,p (I; X), as in the well known case X = R. Lemma 1.4. Let p > 1. If u AC p (I; X) then (the equivalence class of) u W 1,p (I; X). If u W 1,p (I; X) then there exists a unique continuous representative ũ C(I; X) (in particular ũ(t) = u(t) for L 1 -a.e. t I). Moreover ũ AC p (I; X) and the application T : W 1,p (I; X) C(I; X) defined by T u = ũ is a Borel map. Proof. The proof of first assertion can be carried out exactly as in the case X = R (see for example [Bre83] Proposition VIII.3), by using ( h u(t)) p 1 h t+h t u p (r) dr. Now we assume that u W 1,p (I; X) and we consider a sequence {y n } n N dense in X. Defining u n (t) := d(u(t), y n ), the triangular inequality implies u n (t + h) u n (t) d(u(t + h), u(t)). (1.5)

22 Preliminaries The fact that u W 1,p (I; X) and p > 1 implies that u n W 1,p (I) again for [Bre83] Proposition VIII.3. Hence there exist ũ n absolutely continuous such that ũ n = u n a.e. and ũ n is a.e. differentiable. We introduce the negligible set N = n N({t I : ũ n (t) u n (t)} {t I : ũ n(t) do not exists}), and we define m(t) := sup n ũ n(t) for all t I \ N. Clearly, by the density of {y n }, we have for all t, s I \ N, with s < t, d(u(t), u(s)) = sup n ũ n (t) ũ n (s) sup n t s ũ n(r) dr We show that m L p (I). Actually, by (1.5), if t I \ N then t s m(r) dr. (1.6) ũ ũ n (t + h) ũ n (t) n(t) = lim lim inf h h hu(t), (1.7) h which implies m(t) lim inf h h u(t). By Fatou s Lemma and u W 1,p (I; X) we obtain T m(t) p dt C. (1.8) (1.6) and Hölder s inequality show that u : I \ N X is uniformly continuous, thus, by the completeness of X, it admits a unique continuous extension ũ : I X which also satisfies d(ũ(t), ũ(s)) and then, for (1.8), ũ AC p (I; X). t s m(r) dr t, s I, s < t, (1.9) We have thus proved that u AC p (I; X) if and only if u C(I; X) and sup <h<t 1 T ( h u(t)) p dt < +. In order to prove that T is a Borel map, we observe that W 1,p (I; X) and AC p (I; X) are Borel subsets of L p (I; X) and C(I; X) respectively, since the map u T h d p (u(t + h), u(t)) sup <h<t h p dt is lower semi continuous from L p (I; X) to [, + ] and from C(I; X) to [, + ]. Moreover T is an isometry from (W 1,p (I; X), d p ) to (C(I; X), d p ) and the thesis follows by observing that the Borel sets of (C(I; X), d ) coincides with the Borel sets of (C(I; X), d p ). This last assertion is a general fact: if Y is a separable and complete metric space (Polish space) and Y w is the same space with an Hausdorff topology weaker than the original, then the Borel sets of Y coincides with the Y w ones (see for instance [Sch73] Corollary 2, pag 11).

23 1.4 - Borel probability measures, narrow topology and tightness Borel probability measures, narrow topology and tightness This Section contains the definition of narrow convergence for sequences of Borel probability measures in a separable metric space and the important Prokhorov criterion of compactness. Given a separable metric space Y, we denote with P(Y ) the set of Borel probability measures on Y. We say that a sequence µ n P(Y ) narrowly converges to µ P(Y ) if lim ϕ(y) dµ n (y) = ϕ(y) dµ(y) ϕ C b (Y ), (1.1) n + Y Y where C b (Y ) is the space of continuous bounded real functions defined on Y. It is well known that the narrow convergence is induced by a distance on P(Y ) (see [AGS5]) and we call narrow topology the topology induced by this distance. In particular the compact subsets of P(Y ) coincides with sequentially compact subsets of P(Y ). We also recall that if µ n P(Y ) narrowly converges to µ P(Y ) and ϕ : Y (, + ] is a lower semi continuous function bounded from below, then lim inf ϕ(y) dµ n (y) ϕ(y) dµ(y). (1.11) n + Y Y Remark 1.5. As showed in [AGS5], (1.1) can be checked only on bounded Lipschitz functions. A subset T P(Y ) is said to be tight if ε > K ε Y compact : µ(y \ K ε ) < ε µ T, (1.12) or, equivalently, if there exists a function ϕ : Y [, + ] with compact sublevels λ c (ϕ) := {y Y : ϕ(y) c}, such that sup ϕ(y) dµ(y) < +. (1.13) µ T Y The importance of tight sets is due to the following Theorem: Theorem 1.6 (Prokhorov). Let Y be a separable and complete metric space. T P(Y ) is tight if and only if it is relatively compact in P(Y ) Push forward of measures In this Subsection we introduce the push forward operator for probability measures. If Y, Z are separable metric spaces, µ P(Y ) and F : Y Z is a Borel map, the push forward of µ through F, denoted by F # µ P(Z), is defined as follows: F # µ(b) := µ(f 1 (B)) B B(Z), (1.14)

24 2 1 - Preliminaries where B(Z) is the family of Borel subsets of Z. It is not difficult to check that this definition is equivalent to ϕ(z) d(f # µ)(z) = ϕ(f (y)) dµ(y) (1.15) Z Y for every bounded Borel function ϕ : Z R. More generally (1.15) holds for every F # µ- integrable function ϕ : Z R. We will often use this fact. We recall the following composition rule: (G F ) # µ = G # (F # µ) µ P(Y ), F : Y Z, G : Z W Borel functions, and the continuity property: (1.16) F : Y Z is continuous = F # : P(Y ) P(Z) is narrowly continuous. (1.17) Convergence in measure This Subsection is devoted to the link between convergence in measure for maps and narrow convergence of the measures associated to the maps, as in the theory of Young measures. This link is a useful tool in the proof of a stability Theorem in Section 5.1. Let X, Y be metric spaces and µ P(X). A sequence of Borel maps f n : X Y is said to converge in µ-measure to f : X Y if lim µ ({x X : d Y (f n (x), f(x)) > δ}) = δ >. n This is equivalent to the L 1 (µ) convergence to of the maps 1 d Y (f n, f). It is also well known that if Y = R and f n p is equi-integrable, then f n f in µ-measure if and only if f n f in L p (µ). The following Lemma will be used in the proof of Theorem 5.1. Lemma 1.7. Let f n, f : X Y be Borel maps and let µ P(X). µ-measure if and only if (i, f n ) # µ converges to (i, f) # µ narrowly in P(X Y ). Proof. Then f n f in If f n f in µ-measure then ϕ(x, f n (x)) converges in L 1 (µ) to ϕ(x, f(x)), and therefore thanks to (1.15) we immediately obtain the convergence of the push-forward. Conversely, let δ > and, for any ε >, let w C b (X; Y ) such that µ({f w}) ε. We define and notice that X Y ϕ(x, y) := 1 d Y (y, w(x)) δ ϕ d(i, f n ) # µ µ({d Y (w, f n ) > δ}), C b (X Y ) X Y ϕ d(i, f) # µ µ({w f}).

25 1.4 - Borel probability measures, narrow topology and tightness 21 Taking into account the narrow convergence of the push-forward we obtain that lim sup n µ({d Y (f, f n ) > δ}) lim sup µ({d Y (w, f n ) > δ}) + µ({w f}) 2µ({w f}) 2ε n and since ε is arbitrary the proof is achieved. Lemma 1.8. Let f : X Y be a Borel map, µ P(X), and let v L p (µ; R d ) for some p (1, + ). Then, setting ν = f # µ, we have f # (vµ) = wν for some w L p (ν; R d ) with w Lp (ν;r d ) v Lp (µ;r d ). (1.18) In case of equality we have v = w f µ-a.e. in X. Proof. Let q be the dual exponent of p, ν := f # (vµ), and ϕ L (Y ; R d ); denoting by ν α, α = 1,, d, the components of ν we have d α=1 Y ϕ α dν α d = α=1 X (ϕ α f) v α dµ ϕ f L q (µ;r ) v d Lp (µ;r d ) = ϕ Lq (ν;r ) v d Lp (µ;r ). d Since ϕ is arbitrary this proves (1.18) and, as a consequence, the same identities above hold when ϕ L q (ν; R d ). In case of equality it suffices to choose ϕ = w p 2 w to obtain that v coincides with ϕ f q 2 (ϕ f) = w f µ-a.e. in X Disintegration theorem In this Subsection we recall the so-called disintegration theorem for probability measures. It is a classical result in probability theory. The proof can be found, for instance, in [DM78] and in [AFP]. This Theorem is an important tool in Chapters 3 and 4. Let X, Y be separable metric spaces. A measure valued map y Y µ y P(X) is said to be a Borel map if y µ y (B) is a Borel map for any Borel set B B(X). Moreover a monotone class argument implies that y Y f(x) dµ y (x) is a Borel map for every Borel map f : X [, + ]. X Theorem 1.9. Let X, Y be Polish spaces, µ P(X), let π : X Y be a Borel map and ν := π # µ P(Y ). Then there exists a ν-a.e. uniquely determined Borel family of probability measures {µ y } y Y P(X) such that µ y is concentrated on π 1 (y) for ν-a.e. y Y, and X f(x) dµ(x) = for any Borel map f : X [, + ]. Y ( X ) f(x)dµ y (x) dν(y) (1.19)

26 Preliminaries 1.5 Kantorovitch-Rubinstein-Wasserstein distance In this Section we give the definition of Wasserstein distance (more precisely Kantorovitch- Rubinstein-Wasserstein distance) between Borel probability measures on a metric space. Let (X, d) be a separable and complete metric space. We fix p 1 and denote by P p (X) the space of Borel probability measures having finite p-moment, i.e. { } P p (X) = µ P(X) : d p (x, x ) dµ(x) < +, (1.2) X where x X is an arbitrary point of X (clearly this definition does not depend on the choice of x ). Notice also that this condition is always satisfied if the diameter of X is finite; in this case P p (X) = P(X). Given µ, ν P(X) we define the set of admissible plans Γ(µ, ν) as follows: Γ(µ, ν) := {γ P(X X) : π 1 #γ = µ, π 2 #γ = ν}, where π 1 (x, y) := x and π 2 (x, y) := y are the projections on the first and the second component respectively. The p-kantorovitch-rubinstein-wasserstein distance between µ, ν P p (X) is defined by W p (µ, ν) := ( { }) 1 min d p p (x, y) dγ(x, y) : γ Γ(µ, ν). (1.21) X X Since Γ(µ, ν) is tight and µ ν Γ(µ, ν) satisfies X X dp (x, y) dµ ν(x, y) < +, the existence of the minimum, in the above definition, is a consequence of standard Direct Methods in Calculus of Variations. We denote by Γ o (µ, ν) := the set of optimal plans. { } γ Γ(µ, ν) : d p (x, y) dγ(x, y) = Wp p (µ, ν) X X Being X separable and complete, P p (X), endowed with the distance W p, is a separable and complete metric space too. We refer to [Vil3] and [AGS5] for the proof and for a characterization of convergence and compactness in P p (X). Here we only recall that, given a sequence µ n P p (X) and µ P p (X), lim W p(µ n, µ) = n µ n narrowly converge to µ, d p (x, x ) dµ n (x) = lim n X X d p (x, x ) dµ(x). (1.22) We conclude this section by recalling a few basic facts from the theory of optimal transportation (see for instance [GM96], [Vil3], [Eva99], [AGS5] for much more on this subject). In the case when µ P r p(r n ), the subset of P p (R n ) made of absolutely continuous measures with respect to Lebesgue measure L n, it can be shown [Bre91, GM96] that the